. Quasi-smooth hypersurfaces or more generally quasi-smooth intersection sub-\n\nExample 3.2 . Quasi-smooth hypersurfaces or more generally quasi-smooth intersection sub- varieties are quasi-smooth subvarieties (see [2] or [7] for more details).\n\nRemark 3.3 . Quasi-smooth subvarieties are suborbifolds of P d Σ in the sense of Satake in [8]. Intuitively speaking they are subvarieties whose only singularities come from the ambient\n\nProof. From the exponential short exact sequence\n\nwe have a long exact sequence in cohomology\n\nH 1 (O ∗ X ) → H 2 ( X, Z ) → H 2 (O X ) ≃ H 0 , 2 ( X )\n\nwhere the last isomorphisms is due to Steenbrink in [9]. Now, it is enough to prove the commutativity of the next diagram\n\nwhere the last isomorphisms is due to Steenbrink in [9]. Now,\n\nH 2 ( X, Z ) / / H 2 ( X, O X ) ≃